Could PI have a different value in a different universe?

The value of pi is determined by the circumference of a circle.

Why is it any particular constant number? Would a circle as defined as a perfect circle in any universe lead to a different value of pi?

Would all universes where a circle could be constructed by "people" there also lead to the value of pi?

If it is true then it leads to the conclusion that pi is some sort of constant value constant to all universe. What is the meaning of that?

Science fiction references.

In science fiction pi sometimes has a different value in different universes, for example Greg Bear's "The Way", it says "Gates are capped with cupolas formed from Space-time itself. As distortions in space-time geometry, their nature can be calculated by 21st century instruments laid on their 'surfaces'. The constant pi, in particular, is most strongly affected.".

A message is found encoded within pi, in the novel by Carl Sagan, "Contact" "Ellie, acting upon a suggestion by the senders of the message, works on a program which computes the digits of pi to record lengths in different bases. Very far from the decimal point (1020) and in base 11, it finds that a special pattern does exist when the numbers stop varying randomly and start producing 1s and 0s in a very long string.".

Physically, the ratio of a circle's circumference to its diameter $C/d$ is not really $\pi$. General relativity describes gravity in terms of the curvature of spacetime, and roughly speaking, if you take $(C/d-\pi)/A$, where $A$ is the circle's area, what you get is a measure of curvature called the Ricci scalar.

But even if you're doing general relativity, you don't just go around redefining $\pi$. The thing is, $\pi$ occurs in all kinds of contexts, not just as $C/d$. For instance, you could define $\pi$ as $4-4/3+4/5-4/7+\ldots$, which has nothing to do with the curvature of space.

So if you define $\pi$ as $C/d$, you don't even get a consistent value within our own universe, whereas if you define it as $4-4/3+4/5-4/7+\ldots$, you get an answer that is guaranteed to be the same in any other universe.

Another way of looking at it is that $\pi$ is not the $C/d$ ratio of a physical circle, it's the $C/d$ ratio of a mathematically idealized circle that exists in certain axiomatic systems, such as Euclidean geometry. Viewed this way, it doesn't matter that our universe isn't actually Euclidean.

This is a complement of other answers. One can define a value $\pi_p$ as $\pi$ in $ \ell_p$. $\ell_p$ is two dimensional space with a metric as follows: $$ d\left((x_1,y_1),(x_2,y_2)\right)=\left(|x_1-x_2|^p+|y_1-y_2|^p\right)^{1/p} $$ for $1\leq p \leq \infty$. Then the circle $C_p$ is defined as all points $(x,y)$ such that: $$ \left(|x|^p+|y|^p\right)^{1/p}=1 $$ The diameter of this circle is $2$. And therefore we can define $\pi_p$ as the half of circumference of the circle. It can be seen that $\pi_1=4$, $\pi_2=\pi$ and $\pi_\infty=4$. In following image, you can see the value $\pi_p$ versus p:

Reference: Look at the following article $\pi _{p} $ the Value of π in $\ell _p $.

That there will be a number $\pi$ is a mathematical fact. But whether the significant number would be the same is a more interesting question. Some people in our own universe would prefer that the constant had been chosen to be $2\pi$ i.e. $6.28 ...$ instead of $3.14 ... $ as it would reduce the number of factors of 2 in some formulae.

It would also be possible to imagine, in a higher dimensional universe, that the basic round object might be, say, a 3-sphere, with the significant constant would be defined in relation to its geometry rather than the geometry of a circle.

Living in a world which was non-euclidian (e.g. on the surface of a sphere) would make other numbers geometrically significant, but there would still be $\pi$ = $3.14 ...$ sitting in the background.

When I think of different 'universes,' I imagine places that are fundamentally different than our own. Because pi is just the ratio of the circumference to the diameter, that won't change so long as the behavior of the 'metric of the universe' doesn't change.

But suppose that we considered the 'taxi-cab universe,' where the pertinent metric is the taxicab metric (which I have also called the Manhattan Metric, which is nicely alliterative). In such a universe, a circle looks to us to be a square. But within the metric, a circle with radius 4 would have circumference 32. So taxicab-pi would be 4. How nice and even.

I used that as an example, but really it's still just a mathematical creation. One could more or less analyze many different geometries, topologies, manifolds, etc. And to each might be associated some different way of relating a 'circle' (whatever that may mean) to the metric.

Not surprisingly,

$\pi = 2 \int_{0}^{\pi/2} \,ds $

Now substitute,

$ x = \sin s $

then $ 0 = \sin 0 $ and $ 1 = \sin \frac{\pi}{2} $

$ \frac{dx}{ds} = \cos s = \sqrt{1 - x^2} $

$ \pi = 2 \int_{0}^{1} \frac{ds}{dx}\,dx = 2 \int_{0}^{1} \frac{1}{\sqrt{1 - x^2}}\,dx $

No measuring arc lengths with a tape measure. No concept of rotation. Not a trig function in sight. This or any of many other definite integrals could be your mathematical definition of pi.

If you insist on defining $\pi$ as circumference over diameter, there is a whole lot of mechanics you need to give that meaning. And in curved space, circumference over diameter might well depend on the diameter of the circle